aforesaid three observed apparent places of the comet let the three true places be found in this new plane, as well as the orbit passing through them, and the two areas of the same described between the observation, which call δ and ε; and let τ be the whole time in which the whole area δ + ε should be described.
Then taking C to 1 as A to B; and G to 1 as D to E; and g to 1 as d to e; and γ to 1 as δ to ε; let S be the true time between the first observation and the third; and, observing well the signs + and -, let such numbers m and n be found out as will make 2G - 2C, = mG - mg + nG - nγ; and 2T - 2S = mT - mt + nτ. And if, in the first operation, I represents the inclination of the plane of the trajectory to the plane of the ecliptic, and K the longitude of either node, then I + nQ will be the true inclination of the plane of the trajectory to the plane of the ecliptic, and K + mP the true longitude of the node. And, lastly, if in the first, second, and third operations, the quantities R, r, and ρ, represent the parameters of the trajectory, and the quantities 1⁄L, 1⁄l, 1⁄λ, the transverse diameters of the same, then R + mr - mR + nρ - nR will be the true parameter, and will be the true transverse diameter of the trajectory which the comet describes; and from the transverse diameter given the periodic time of the comet is also given. Q.E.I. But the periodic times of the revolutions of comets, and the transverse diameters of their orbits, cannot be accurately enough determined but by comparing comets together which appear at different times. If, after equal intervals of time, several comets are found to have described the same orbit, we may thence conclude that they are all but one and the same comet revolved in the same orbit; and then from the times of their revolutions the transverse diameters of their orbits will be given, and from those diameters the elliptic orbits themselves will be determined.
To this purpose the trajectories of many comets ought to be computed, supposing those trajectories to be parabolic; for such trajectories will always nearly agree with the phænomena, as appears not only from the parabolic trajectory of the comet of the year 1680, which I compared above with the observations, but likewise from that of the notable comet which appeared in the year 1664 and 1665, and was observed by Hevelius, who, from his own observations, calculated the longitudes and latitudes thereof, though with little accuracy. But from the same observations Dr. Halley did again compute its places; and from those new places determined its trajectory, finding its ascending node in ♊ 21° 13′ 55″; the inclination of the orbit to the plane of the ecliptic 21° 18′ 40″; the distance of its perihelion from the node, estimated in the comet's orbit, 49° 27′ 30″, its perihelion in ♌ 8° 40′ 30″, with heliocentric latitude south
